`int(d(x^(3)))/(x^(3)(x^(n)+1))` equals

Options

A. `(3)/(n)ln((x^(n))/(x^(n)+1))`
B. `(1)/(n)ln((x^(n))/(x^(n)+1))`
C. `(3)/(n)ln((x^(n)+1)/(x^(n)))`
D. `3nln((x^(n+1))/(x^(n)))`

Explanation

To solve the integral \( \int \frac{d(x^3)}{x^3 (x^n + 1)} \), we can follow these steps:

### Step 1: Rewrite the Integral
Start by rewriting the integral in a more manageable form:
\[ \int \frac{d(x^3)}{x^3 (x^n + 1)} = \int \frac{3x^2 \, dx}{x^3 (x^n + 1)} \]

### Step 2: Simplify the Expression
Now simplify the expression:
\[ = \int \frac{3 \, dx}{x (x^n + 1)} \]

### Step 3: Multiply and Divide by \( x^{n-1} \)
Next, multiply and divide the integrand by \( x^{n-1} \):
\[ = \int \frac{3x^{n-1} \, dx}{x^n (x^n + 1)} \]

### Step 4: Change of Variables
Let \( t = x^n \). Then, differentiate both sides:
\[ dt = n x^{n-1} \, dx \quad \Rightarrow \quad dx = \frac{dt}{n x^{n-1}} \]
Substituting this into the integral gives:
\[ = \int \frac{3}{t(t + 1)} \cdot \frac{dt}{n} \]

### Step 5: Factor Out Constants
Factor out the constant \( \frac{3}{n} \):
\[ = \frac{3}{n} \int \frac{1}{t(t + 1)} \, dt \]

### Step 6: Partial Fraction Decomposition
Now, perform partial fraction decomposition on \( \frac{1}{t(t + 1)} \):
\[ \frac{1}{t(t + 1)} = \frac{1}{t} - \frac{1}{t + 1} \]

### Step 7: Integrate
Integrate the expression:
\[ = \frac{3}{n} \left( \int \frac{1}{t} \, dt - \int \frac{1}{t + 1} \, dt \right) \]
\[ = \frac{3}{n} \left( \ln |t| - \ln |t + 1| \right) + C \]

### Step 8: Substitute Back
Substituting back \( t = x^n \):
\[ = \frac{3}{n} \left( \ln |x^n| - \ln |x^n + 1| \right) + C \]
\[ = \frac{3}{n} \ln \left( \frac{x^n}{x^n + 1} \right) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{d(x^3)}{x^3 (x^n + 1)} = \frac{3}{n} \ln \left( \frac{x^n}{x^n + 1} \right) + C \]

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